MONEY AND THE (C)CAPM: THEORY AND EVALUATION

Transcription

1 MONE AND THE C)CAPM: THEOR AND EVALUATION MARCH 2004 Ronald J. Balvers Dayong Huang Division of Economics and Finance Division of Economics and Finance P.O. Box 6025 P.O. Box 6025 West Virginia University West Virginia University Morgantown, WV Morgantown, WV Phone: 304) Phone: 304) ABSTRACT We consider asset pricing in a monetary economy where liquid assets are held to lower transaction costs. The ensuing model extends the CAPM and the Consumption CAPM by deriving real money growth as an additional factor determining returns. Empirically, the unconditional version of this model compares favorably to other theoretical asset pricing models. Allowing for conditional variation in factor sensitivities improves model performance so the model performs as well as the a-theoretical Fama-French three factor model. The paper further introduces a technique that facilitates derivation of dynamic asset pricing results in discrete time by generalizing Stein s Lemma to multivariate cases. JEL classification: G12 Keywords: Asset Pricing, Money Supply Growth, Consumption CAPM, Stein s Lemma

2 Money and the C)CAPM: Theory and Evaluation ABSTRACT We consider asset pricing in a monetary economy where liquid assets are held to lower transaction costs. The ensuing model extends the CAPM and the Consumption CAPM by deriving real money growth as an additional factor determining returns. Empirically, the unconditional version of this model compares favorably to other theoretical asset pricing models. Allowing for conditional variation in factor sensitivities improves model performance so the model performs as well as the a-theoretical Fama-French three factor model. The paper further introduces a technique that facilitates derivation of dynamic asset pricing results in discrete time by generalizing Stein s Lemma to multivariate cases.

3 Money and the C)CAPM: Theory and Evaluation Introduction The empirical performance of the basic asset pricing theories the CAPM and the Consumption CAPM has been disappointing. Alternative theories such as Cochrane s 1996) investment-based model and Lettau and Ludvigson 2001a, 2001b) C)CAPM conditional on the consumption-to-wealth ratio perform considerably better, but the purely-empirical three factor model of Fama and French 1996) still outperforms all theoretical asset pricing formulations. A major challenge to the profession is to provide a set of pricing factors that is both reasonable from a macroeconomic perspective and is competitive with the Fama-French model in pricing the cross section of asset returns. From a macroeconomic perspective, the monetary environment as related to money growth and inflation appears to be a prime candidate for providing a systematic pricing factor. Early empirical results starting with Chen, Roll, and Ross 1986) were suggestive of a role for money in asset pricing by finding that both anticipated inflation and unanticipated inflation are significant factors in pricing the cross-section of assets. Following these results, a large body of work has investigated the link between equity prices and inflation or money growth. Boyle and oung 1988), for instance, consider a static asset pricing model in which individuals have direct utility over consumption as well as real money balances. In their framework, both monetary shocks and real shocks affect the equity premium. Marshall 1992) obtains similar results but employs a dynamic formulation in which the level of liquidity measured by real money holdings affects the cost of doing transactions. Both of these papers focus primarily on the link 2

4 between stock prices and inflation and do not consider cross-sectional pricing implications. Similarly, Boyle and Peterson 1995), Thorbecke 1997), Jensen and Mercer 2002), and others examine the monetary policy impact on asset pricing without considering cross-sectional asset pricing implications. Chan, Foresi and Lang 1996) examine the implications for cross-sectional asset pricing of a liquidity constraint in a cash-in-advance economy in which certain goods can be purchased only with money. In this environment real inside money holdings are equal to the consumption good that requires cash-in-advance. Due to a separability in utility assumption, the stochastic discount factor depends solely on the growth rate of this cash-in-advance consumption good. Hence, the growth rate of real inside money balances becomes the asset pricing factor. They find that their model with real inside money growth as the sole factor outperforms the C-CAPM, but performs marginally worse than the market CAPM. 1 From a different angle, Pástor and Stambaugh 2003) find that aggregate liquidity is a systematic asset pricing factor that moreover can explain a large part of momentum profits. They measure aggregate liquidity as the average price reversals induced by order flow. The concept of liquidity used by Pástor and Stambaugh related to whether financial assets can be traded quickly and at low cost is a micro liquidity concept and quite different from the macro liquidity concept related to the monetary environment, which deals with whether consumption goods can be traded easily at low time and resource cost). Intriguingly, Chordia, Sarkar, and Subrahmanyam 2002) link 1 The money factor, even if it is theoretically equivalent to consumption, can outperform the consumption factor given that consumption services are difficult to measure. A similar motivation has led to the production-based asset pricing models of Balvers, Cosimano, and McDonald 1990) and Cochrane 1991) in which output growth and investment growth, respectively, are merely alternative proxy variables measuring the marginal rate of intertemporal substitution in consumption. 3

5 the two liquidity concepts empirically by showing that improvements in stock market liquidity are tied to nominal) money growth. This paper follows the formulation in Marshall 1992) to introduce money to the C-CAPM in a very natural way. The monetary economy is grounded in a transactions and precautionary demand for money that stems from the liquidity services of money holdings as means to lowering the cost of doing transactions. 2 In this environment, the marginal value of financial returns is determined by the marginal utility of consumption as well as the marginal cost of doing transactions which depends on the liquidity currently available. Accordingly, the stochastic discount factor varies with real consumption growth as well as with real money growth, where the real money growth factor combines the effects of nominal money growth and inflation. Using the envelope condition, it follows that the marginal utility of wealth alternatively depends on real market returns together with real money growth rates. Thus, a two-factor asset pricing model emerges with either market return and real money growth, or consumption growth and real money growth, as the factors. The model solution that yields the asset pricing implications is accomplished in discrete time by introducing a new method based on extending Stein s 1973, 1981) lemma to multivariate applications. The main advantage of this discrete-time approach is that it allows a direct transfer of the theoretical results into an empirical formulation, without worrying about time aggregation. It also permits a setting that may be more intuitive than the continuous-time setting typically employed for analyzing cross-sectional pricing in equilibrium dynamic asset pricing models, and, moreover, facilitates dealing with lags and conditioning information. Following Lettau and Ludvigson 2001b) we employ the Fama-MacBeth 1973) method 2 Marshall s formulation originates with Brock 1974) and Feenstra 1986). See also the description in Walsh 2003, Chapter 3). 4

6 with Shanken and GMM-based corrections for measurement error in the betas) to price the crosssection of the 25 Fama and French portfolios ranked by size and book-to-market ratio. We find that our real money-growth-augmented CAPM explains 65%, and the real money-growth-augmented C- CAPM explains 60% of the cross-sectional returns variation, in either case with insignificant Jensen alphas. Both compare favorably with the CAPM, the C-CAPM, and with the theories of Cochrane 1996), Campbell and Cochrane 1999), and Lettau and Ludvigson 2001a, 2001b). If we allow for conditional variation in the betas, using the standard instrument of the lagged term premium as the conditioning factor, then our model performance improves further. Explaining 75% of the crosssectional returns variation this conditional model performs as well as the a-theoretical Fama-French 1992, 1993, 1996) three-factor model. In Section 1 we derive a discrete-time factor model based on the assumption of conditional normality of returns and factors. We derive both a money CAPM and a money C-CAPM and consider unconditional and conditional specifications of each. Section 2 empirically evaluates each version and compares to alternative asset pricing models. For the sake of robustness and to facilitate comparisons we provide results for alternative conditioning variables, and also conduct alternative evaluations and obtain comparable estimates based on a GMM-stochastic discount factor approach. Section 3 concludes. 1. Asset Pricing in a Monetary Economy We add money to the standard C-CAPM model of Breeden 1979). Money is held because it lowers the transactions cost of purchasing consumption goods, as modeled in Marshall 1992) who, however, did not consider cross-sectional asset pricing issues. The addition of money is best 5

7 modeled in discrete time since money must be acquired first before it can be used to facilitate transactions. Analogously to the Brownian motion assumption in continuous time, we assume conditional normality and are able to obtain a specific solution for the asset pricing equation by providing and applying an extension to Stein s lemma. The resulting model has substantially different implications compared to Chan, Foresi, and Lang 1996) the only previous paper to have considered the theoretical cross-sectional asset pricing implications of money by generating real money balances as an additional factor to consumption instead of an alternative proxy. a) The Model A representative consumer-investor in an endowment economy maximizes expected lifetime utility subject to a budget constraint and given the fact that transactions costs to purchase consumption goods are mitigated by money holdings. The maximized value of life time utility in real terms is given by the Bellman equation of dynamic programming as:, 1) where and represent real financial wealth excluding money holdings) and real money holdings, respectively; and indicates a set of state variables, exogenous to the consumer-investor, that is sufficient to represent changes in the investment opportunity set. The agent in period t chooses current consumption, real money holdings for use in the upcoming period), and the portfolio shares of a risk free asset asset 0) and n risky assets. Life time utility is time separable, 6

8 future utility is discounted with factor, and felicity u.) in each period is a concave function of consumption. Optimization is subject to the budget constraint for each period: with, 2),, 3). 4) represents the real transactions cost of purchasing the current level of consumption. Indicating partial derivatives by subscripts, the conditions on the transactions cost are that,,,,,. These conditions are intuitive. Transaction costs are positive, increase in consumption and decrease in real balances. The second derivative conditions assure that increases in consumption and decreases in real money holdings have a constant effect or become increasingly costly. The cross derivative condition implies that the marginal cost of consumption stays constant or falls as more real money balances are held See Feenstra 1986, p. 278), Marshall 1992, pp ), and Walsh 2003, pp ) for more discussion on the conditions on the transactions cost function. Feenstra 1986) shows that the transactions cost formulation used here can be derived from standard theories of transactions and precautionary demand for money. Low liquidity implies, for instance more trips to the bank and more costs of having to buy goods on credit. He also shows that the transactions cost approach is equivalent to the Sidrauski assumption of putting money directly in the utility function. Further, the popular cash-in-advance approach is just a limiting case of this approach when becomes infinite for. Thus, the model employs a quite general approach to incorporating money. 4 To assure that second-order conditions for the maximization problem are met, we need to have a concave objective and a convex constraint set. This is guaranteed by the concavity of the utility function and the conditions on the transaction cost function see for instance Lucas and Stokey 1989, Chapter 4). 7

9 Investable wealth in equation 2) equals and future money holdings minus current expenditure on consumption. The gross real portfolio return is a weighted average of returns on the individual assets as given by equation 3). Changes in the distribution over time of the asset returns are captured by the state variables the Merton, 1973, factors). The choice of real money balances at the current price level to be used in the next period differs from the real balances actually used in the next period at the then relevant price level, due to the gross inflation rate money creation, and due to an assumed transfer of government revenues from nominal, to the representative consumer-investor, yielding equation of motion 4). The first-order conditions derived from 1) - 4) are:, 5), 6). 7) The envelope conditions for the two state variables are:, 8). 9) 8

10 From aggregating equation 7) multiplied by the portfolio shares over all i and using equation 8) it follows that:. 10) Defining excess returns as, equation 7) becomes, 7') yielding the typical result that we have the marginal value of wealth serve as the stochastic discount factor pricing all assets. However, it follows from equations 5) and 8) that, 11) which is non-standard in that the marginal utility of consumption is not sufficient to capture the marginal value of wealth. The marginal value of wealth depends on consumption but also on the real value of money holdings, determining the cost of purchasing consumption goods at the margin. Given consumption, if money holdings turn out to be low after the price level and market returns are realized a liquidity squeeze), then the marginal cost of consumption is higher at this given consumption level so that the marginal value of wealth is lower according to equation 11). We continue in the spirit of Merton 1973) and Breeden 1979) by developing an equilibrium 9

11 asset pricing model without formally specifying a full general equilibrium environment. 5 To isolate the contribution of the money factor, we rule out Merton factors by assuming that there are no changes over time in the exogenous dividend processes, ruling out shifts in the investment opportunities set except, possibly, through changes in the real money supply). 6 Then we have. 12) We further assume that all asset returns and the factors are conditionally multivariate normal. 7 This is an assumption made in many equilibrium asset pricing models. Even if normality does not hold exactly, the analysis is appropriate if one is willing to accept the assumption, bolstered by various Central Limit theorems, that a variety of different shocks adds up to an approximate normal distribution. Working in discrete time with conditionally normal distributions requires the following lemma. b) A Multivariate Version of Stein s Lemma Based on equation 7') we derive a two factor model under the assumption of conditional 5 A general equilibrium version of the model embedded in a Lucas 1978) endowment economy is available from the authors. Here dividend processes and nominal money growth are the exogenous endowment processes and Merton factors can be ruled out if dividends and money growth are i.i.d. Even if the endowment processes follow conditionally normal processes, multivariate conditional normality of asset returns and the factors, as we assume, is not guaranteed in general but holds for some examples in general equilibrium. 6 If the distributions of returns are not predictable then investment opportunities do not change, but the realized level of real money balances still affects return realizations in our model. Thus, even if equilibrium real money balances are not predictable as in many newclassical macro models), they may still be a priced risk factor. 7 In fact the solution technique only requires that the factors are functions of conditionally normal random variables but for simplicity we avoid this slight generalization. Conditional normality can imply a large variety of unconditional return distributions, analogously to the continuous-time formulation where the Brownian motion building blocks can imply other return distributions when aggregated over time. 10

12 normality. We assume that the returns of all assets as well as real money growth are conditionally multivariate normal. To be able to derive a multi-factor model we must first provide a generalization of Stein s Lemma which has not appeared elsewhere in the finance or statistics literature. Multivariate Extension of Stein s Lemma: If X,, and Z are multivariate normal and h ) is differentiable, with E h, Z ) < for, then i. 13) Proof: See Appendix. Note that the statement of the lemma here is specialized to a trivariate normal case, which is the case we need in this paper, but that the general multivariate case and its proof are provided in the appendix. c) Derivation of the Money CAPM The derivation of the asset pricing equations extends a standard approach for single-factor models see for instance Huang and Litzenberger, 1988) to multi-factor models. Use the definition of covariance in equation 7') and employ equation 12) to dispense with the Merton variables:. 14) Apply the multivariate version of Stein s Lemma in equation 13) to equation 14), noting from 11

13 equation 2) and our normality assumptions that is conditionally normal, to produce, 15) with. 16) Here double subscripts indicate second derivatives. Assume that an asset exists with a payoff that moves perfectly with real money growth. Such an asset, has excess return, 17) where and is positive and known at time t. 8 Relating the covariances in equation 15) to the market excess return and to the real money supply growth rate gives, 18) where, from equation 2) and the definition of, as 8 From equation 10) and the definition of return, this asset would have a price of:. Similarly, we have the price of a riskless asset with unit payoff) given. Accordingly, an asset position m exists with excess return for which. 12

14 ,, 19) Based on equations 5) - 9) it can be shown that confirming risk aversion) and that the sign of is ambiguous. 9 Thus, from equation 16), and 0. Applying equation 18) to the market asset and asset m implies:, 20). 21) If the sign of is positive, then < 0 from equations 16 and 19). In this case, an asset with return positively correlated with money growth has a lower expected return as follows from equation 18). The money growth factor then is a hedging factor in the sense that positive exposure of some portfolio to money growth mitigates risk since this portfolio tends to pay out well when the marginal value of wealth is high when > 0). Even if < 0, from equation 21) the expected excess return on the asset whose return is perfectly correlated with money growth,, may be positive or negative. The reason is that a positive hedging benefit may be offset by positive exposure to market risk if the covariance between market return and money 9 The sign of can be obtained by differentiating in equation 11) with respect to m, using the fact that optimal consumption is determined by the state variables,. The effect of higher money holdings is to raise the marginal value of wealth as the marginal transactions cost falls given ). However, higher money holdings also have a price effect in that consumption becomes cheaper due to lower transactions cost. This lowers the marginal value of wealth as the higher optimal consumption level reduces the marginal utility of consumption and increases the marginal transactions cost. 13

15 growth is positive. 10 Clearly, if # 0 then the money growth risk premium must be positive given that the covariance between market return and money growth is positive. Solving for and from equations 20) and 21):, 22) and plugging into equation 18) yields, 23) where, 24). 25) Here the coefficients are analogous to those obtained in a multivariate regression. Equation 23) thus provides a two-factor model of asset returns. The two factors are the observable 10 In our data, the is in fact positive, equal to Similarly, and. 14

16 market excess return and the money growth rate minus an unobservable variable that is constant across assets. There is a standard market beta and a real money growth gamma. For a positive money factor risk premium,, if asset i has positive exposure to money growth risk,, then asset i s expected return is higher since the money factor exacerbates risk. Intuitively, when liquidity is high so that transactions are cheap and accordingly optimal consumption is high lower marginal utility of consumption), the extra wealth from a higher return on asset i positively correlated with liquidity tends to be less useful. Asset i is thus less attractive so that a higher expected return is required. d) Derivation of the Money C-CAPM Instead of using the marginal value of wealth as the stochastic discount factor, based on the envelope condition provided in equation 11), we can directly employ the transaction-cost adjusted marginal utility of consumption condition instead to derive a C-CAPM with money growth as an additional factor. An analogous derivation to that of the Money CAPM in equation 23) yields with, 26), 27). 28) 15

17 Here represents the consumption growth rate and a positive constant known at time t. The signs of both risk premia are ambiguous a priori. The money growth factor matters to deal with the risk of changes in transactions cost. These transactions cost have both an income and a substitution effect. The substitution effect occurs because a lower cost of doing transactions implies higher optimal consumption lower marginal utility). So an asset that pays out mostly when liquidity is high is less valuable and requires a higher return). The confounding income effect occurs for given planned consumption: an asset that pays out most when liquidity is high is worth more and requires a lower return) because it allows more transactions exactly when transactions cost are low. The risk premium on the money growth factor is positive if the substitution effect dominates the income effect, but this cannot be guaranteed a priori. e) From Theory to Measurement Unconditional models Under the same general equilibrium assumptions that rule out Merton factors, that nominal money growth and dividend processes are i.i.d., the betas are constant over time. Taking unconditional expectations, equation 23) then becomes. 29) Similarly, equation 26) becomes: 30) 16

18 These represent our unconditional versions of the models to be estimated. The standard Fama- MacBeth 1973) two-pass beta approach under the assumption that the betas are stationary provides consistent estimates of the coefficients of each asset in the first pass. In the second pass, then, the specifications in equations 29) and 30) make clear that the constant should be zero asymptotically only due to measurement error in the generated regressors) if the estimates of the coefficients are used as regressors. Standard errors in tests of significance use the Shanken correction to adjust for measurement error. As discussed in the previous section, the signs of the risk premia to be obtained in the second pass are ambiguous, except for the sign of which should be positive because average market excess returns are positive in the data. One other restriction is that the risk premium for the money factor,, should be identical in both the Monetary CAPM and the Monetary C-CAPM versions. Equation 29) can be used to derive straightforwardly a stochastic discount factor see Cochrane, 2001, section 6.3):. 31) This factor is the unique stochastic discount factor that can be mimicked by a portfolio of existing assets. Although it should price all assets just as well as the stochastic discount factor in equation 10),, the latter is generally much different since it need not be normally distributed and then, given our normality assumption, cannot be generated by a portfolio of existing assets. Conditional models 17

19 If investment opportunities vary over time, when the endowment shocks are not i.i.d., the factor loadings generally covary with the factor premia. To deal with this issue, we derive the mimicking stochastic discount factor directly from the conditional formulation in equation 23). This yields. 32) We allow the factor loadings to vary linearly with a single conditioning variable, again as in Lettau and Ludvigson 2001b), and follow Cochrane 1996) and others in utilizing the term premium at each time as a conditioning variable. Then:. 33) This stochastic discount rate formulation can be transformed back into a beta model, which becomes after taking unconditional expectations:. 34) Similarly, from equation 26) we can derive the conditional Money C-CAPM as:. 35) 18

20 Equations 34) and 35) are our formulation of the conditional versions of the Money CAPM and the Money C-CAPM. In the following we focus on estimating stochastic versions of equations 29), 30), 34), and 35) and comparing the results to alternative asset pricing models. 2. Empirical Results and Comparison to Alternative Asset Pricing Models In this section we discuss our methodology, the data, the results for both unconditional and conditional models, and alternative evaluation methods. a) Methodology We employ the Fama-MacBeth 1973) two-pass beta approach to estimate and evaluate the various asset pricing models. Under the assumption of beta stationarity it is optimal to estimate betas using the full sample. In this case, Cochrane 2001) shows that coefficient estimates from running a cross-sectional second pass are identical to those from averaging the coefficient estimates of the cross-sectional regressions for each time period. We use the latter approach but making Shanken 1992) and GMM corrections to the t-statistics to account for measurement error in the generated regressors the betas) and to provide results robust to heteroskedasticity and serial correlation in the errors. 11 The beta approach under beta stationarity is also equivalent to a stochastic discount factor approach with identity weighting matrix in the GMM estimation. We follow Lettau and Ludvigson 11 As Jagannathan and Wang 1998) point out concerning the measurement error issue, the Fama-MacBeth method need not overestimate the precision of the standard errors in the presence of heteroskedasticity. We thus provide the Fama-MacBeth t-values as well as the Shanken t-values. Cochrane 2001) provides a convenient GMM approach for calculating the t-statistics, which we apply here with five lags see John Cochrane s website for details). These GMM t-statistics impove on the Shanken adjusted t-values in the presence of heteroskedastic or autocorrelated errors. 19

21 in not conducting second-stage GMM estimation with an optimal weighting matrix. The reasons are that estimation of the weights is poor when the time series is short relative to the cross section. This is the case here since we have quarterly data while working with 25 test assets. Further, since the 25 Fama-French portfolios are chosen as test assets because of their economic relevance, focusing on extreme positions on these assets, as the optimal weighting is likely to do, may not be desirable. Lastly, as the optimal weights on the different test assets differ from model to model, comparison across models is more difficult. We also consider the Hansen-Jagannathan weights for model evaluation although for our purposes these share some of the shortcomings of the optimal weights) because they are constant across the models and therefore allow comparison. For further discussion on these issues see Lettau and Ludvigson 2001b, p. 1276) and Cochrane 2001, Ch. 11). The beta approach allows evaluation of the results relying on two general criteria: First, the significance of the coefficient estimates the risk premia) relative to their theoretical values. The theory implies a) an intercept of zero. As our derivations show, this holds even if the factors are not returns. Additionally, the theory may suggest b) the sign of a risk premium or the difference between the actual value of the risk premium and the average return on the factor if the factor is a return). In a one-factor model such as the CAPM, implications a) and b) are perfectly correlated in that a deviation from a) implies a deviation from b) and vice versa. Since, most of the signs of the risk premia are ambiguous in our model, we focus on a), checking the intercept. Note that this requires that we consider returns net of the return on a risk free asset. In this we differ from Lettau and Ludvigson 2001b) since they assume that no risk free asset exists. The second evaluation criterion deals with the fit of the model in explaining the returns of the set of test assets. The theory implies that the stochastic discount factor should price all assets. 20

22 In principle, considering equations 29) and 30) for instance, the average returns should be explained perfectly by the factor sensitivities. Thus, a) a higher R-square suggests a better model. Further, b) additional factors should have no significant impact in explaining the cross-sectional returns. Accordingly, in evaluating our models and comparing against the standard alternative models we focus on the significance of the intercept, Jensen s alpha; the adjusted R-squared measure of the goodness of fit as well as the average pricing error; and the influence of additional factors. b) The Data As the test assets we employ the time series of the returns on the 25 portfolios sorted by size and book-to-market ratio which are provided by Fama and French available from Kenneth French s website). These are the assets that are most commonly used to test asset pricing models and present the greatest challenge to asset pricing theory. Since we consider excess returns, we cannot use the risk free asset as a 26th test asset. The three month t-bill rate is used as the risk free asset. Nominal money growth is based on the time series of M2 and is deflated by the Consumer Price Index, consumption is proxied by quarterly non-durables consumption. The term premium is constructed as the difference between ten-year government bonds and the three-month T-bill rate. These series are available from the Federal Reserve Bank of St. Louis. Market returns are the value-weighted returns index from CRSP. Due to the quarterly availability of the consumption data, all series are compounded from monthly to quarterly. 12 The availability of M2 data further limits our sample to 12 Monthly consumption growth data are available and used by Hodrick and Zhang 2001). However, these data are unreliable as a measure of the stream of consumption services. In particular, they exhibit significant measurement error as evidenced by a significant first-order correlation coefficient of Accordingly, we follow the previous literature e.g., Lettau and Ludvigson, 2001a, 2001b) in employing quarterly consumption data. 21

23 start in 1959Q1. The sample ends with 2003Q1. 13 c) Results for Unconditional Model Specifications Table 1 provides the results for our unconditional models: 1) the Money CAPM and 2) the Money C-CAPM as well as unconditional alternatives: 3) the CAPM, 4) the C-CAPM, 5) the Fama-French 1996) three factor model, and 6) Cochrane s 1996) investment-based model. The Money CAPM yields a good fit with adjusted R-squared of 65%, positive risk premia for market return and the real money growth rate with the latter being strongly significant even using the Shanken t-statistic. The intercept implies a risk-adjusted return Jensen s alpha that is insignificant and equals 0.90 % per quarter. The Money C-CAPM yields an R-squared of 60% and an insignificant intercept of 1.07 % per quarter. Thus, results for both versions of the model are similar as would be expected given the model which implies that both equations 29 and 30 are correct) if consumption growth is measured about as accurately as wealth growth/the market return. 14 As is well known the regular CAPM and C-CAPM both perform poorly with adjusted R- squared around 6%, a significant intercept implying a 3% risk-adjusted return per quarter, and wrong signs for the factors. The adjusted R-squared for Cochrane s investment-based model is 21% with a statistically insignificant) risk-adjusted return of 1.92% per quarter. The Fama-French three 13 Other data used for evaluating competing models are the cay ratio from Sydney Ludvigson s web site, the book-to-market and size variables average of logs over time) from Kenneth French s website, the investment growth data non residential and residential) from the St. Louis Fed website, the dividend yield from CRSP. 14 Where there are definite predictions, the signs of the risk premia are consistent with the theory. First, as shown in Table 1, the risk premia for the money factor are not significantly different in the two models; and, second, the market return risk premium is positive, although not significant. 22

24 factor model has adjusted R-squared of 75% but has a risk-adjusted return of 3.17% per quarter which is statistically significant. 15 Our Money CAPM and Money C-CAPM both seem to perform quite well unconditionally; we do not observe the puzzling performance that the CAPM and C-CAPM present. Figure 1 illustrates the fit for each of the six unconditional models by showing the deviations of each test asset s return from that predicted by the model. For our models, the pricing errors are largest for the 11 and 41 portfolios smallest size quintile and smallest book-to-market ratio, and smallest size quintile and 4 th smallest book-to-market ratio, respectively) which is similar for the Fama- French three factor model. d) Results for Conditional Model Specifications Table 2 provides the results for six conditional models: 1) the Money CAPM and 2) the Money C-CAPM with the term premium as the conditioning factor, 3) the C-CAPM with the term premium as the conditioning factor, 4) the Market Cay and 5) the Consumption Cay models of Lettau and Ludvigson 2001a, 2001b), and 6) the Habit-Based Consumption CAPM of Campbell and Cochrane 1999) and Li 2001). Each of the conditional models have insignificant intercepts. The conditional Money CAPM and Money C-CAPM formulations generate adjusted R-squared values of 75% and 70%, 15 Since the previous literature e.g., Chen, Roll, and Ross, 1986, and Flannery and Protopapadakis, 2002) has illustrated the importance of a variety of macro variables, we also considered various macro variables as factors such as real GNP, the risk free rate, inflation, and nominal money growth. Adding any of these factors to consumption growth or the market risk premium yields results that are significantly inferior to our benchmark results, with R-squared values varying from 15% to 48%. When we add both nominal money growth and inflation separately to the consumption growth or the market risk premium model, the risk premia on these factors are of opposite sign and similar in absolute value, as predicted by our model, but a formal test narrowly rejects the restriction that the absolute values are equal pvalue of 0.04 in both cases). These results are available from the authors. 23

25 respectively. The C-CAPM using the term premium as a conditioning factor and the Habit-Based Consumption model, which to our best knowledge has not been subjected previously to crosssectional testing, each yield an R-squared of 62%. Both Cay models have adjusted R-squared of around 54%. These results are depicted in Figure 2. e) Further Evaluation Size and book-to-market variables In order to test for irrelevant factors, we include the size and book-to-market variables as additional factors in the second pass as suggested by Jagannathan and Wang 1998). Table 3, panel A provides the results for the size factor and panel B provides the results for the book-tomarket factor added individually to our four specifications, the two Cay models, and the Fama- French three-factor model. Each of the models is rejected based on one or both of these added variables. 16 This includes also the Fama-French three factor model. The blanket rejection of all models along this dimension suggests a further search for a specification that more fully absorbs the size and value effects, and implies that our money-modified CAPM is not the full story as, in the end, no model can ever be). In particular, the fact that our real money factor and the Fama-French factors independently add explanatory power suggests that the view of the value and size factors as proxies for liquidity risk may be only a part of the explanation for the importance of these factors. Beta-approach pricing errors Table 4 presents the squared pricing errors for the book-to-market and size quintiles for all 16 Although Table 3 does not display the results for all the models, each of the 12 models that we consider is rejected based on either the size or the book-to-market variable. 24

26 models. Ranking of each model by its average pricing error is equivalent to that implied by ranking the adjusted R-squared values as pointed out by Lettau and Ludvigson 2001b). Evaluation based on the P 2 -statistic is somewhat different since besides the pricing errors both the covariance matrix of the factors and of the pricing errors are involved. Cochrane s investment growth model outperforms all other unconditional models from this perspective. However, none of our four models can be rejected based on this test. GMM distance tests An alternative model evaluation method measures the pricing errors when the expected returns are based on each model s stochastic discount factor the pricing kernel). The distance between each model s stochastic discount factor and a true stochastic discount factor is given by the Hansen-Jagannathan HJ) distance. This measure is obtained as the square root of the minimized Generalized Method of Moments GMM) objective when we use the second moment matrix of asset returns to weight the moment conditions; it is appropriate for comparing models as the weights do not depend on model characteristics. The HJ-distance is reported in Table 5 for each of the 12 models together with p-values for the null-hypothesis that the distance is zero. None of these models survive the HJ test. Ahn and Gadarowski 2004) point out however that the HJ distance causes the correct model to be rejected too often for commonly used sample sizes. As recommended by Altonji and Segal 1996), Cochrane 1996, 2001), and Lettau and Ludvigson 2001b), we compute an alternative distance measure using the identity matrix instead of the HJ weighting matrix. In this case, the distance is given by the square root of the minimized GMM objective using the identity matrix as the weighting matrix. The distance results for the 25

27 identity weighting matrix are also reported in Table 5. The six models with the smallest distance are our four models plus the Fama-French model and the Campbell-Cochrane Habit C-CAPM. While the distance measures are not significantly different from zero for all of the conditional models as well as our two unconditional models, the Fama-French model is rejected based on the distance with identity weighting matrix. Lettau and Ludvigson 2001b) also report rejecting the Fama-French model based on this measure and point out that this may be in part because models with returns as factors have smaller sampling errors in the estimated betas. Note that, as expected from the equivalence of the Fama MacBeth method and first-stage GMM method with identity matrix weights, the p-values reported here are consistent with the joint test of pricing errors results reported in Table Other conditioning variables We do not consider other conditioning variables jointly with the term premium, because each conditioning variable adds additional factors equal to the number of unconditional factors to the model. To check robustness we instead try alternative proxies for changes in investment opportunity sets individually: the risk free rate suggested originally by Merton, 1970) and the dividend yield typically used as a conditioning variable in the asset pricing context see for instance Cochrane, 1996). Table 6 shows that the results for both our MCAPM and the MC-CAPM are at least as good with R-square ranging from 78% to 89%) when the risk free rate or the divided yield is used as the conditioning variable instead of the term premium. 17 To check for consistency we also obtain via one-stage GMM the stochastic discount factor coefficients for each unconditional model and calculate the implied risk premia and p-values via the delta method following Cochrane, 2001, and Hodrick and Zhang, 2001). The results are very close to those in Table 1 and are not displayed. 26

28 3. Conclusion The promise of the consumption-based asset pricing model of Breeden 1979) that a consumption growth factor alone is adequate for pricing any asset is based on the fact that the consumption level is a sufficient statistic for capturing marginal utility of the representative consumer, or any consumer in complete financial markets. However, monetary theory suggests that a further factor is relevant. Specifically, liquidity as determined by the real quantity of money balances, affects the cost of acquiring consumption goods and thus affects the transactions-costadjusted marginal utility of consumption. A generalized version of Stein s Lemma permits the derivation of a two-factor model, the Money CAPM, or, using the envelope condition, an alternative two-factor model, the Money C-CAPM. This two-factor model, in both versions, is subjected to the data. Both model versions perform about equally well and outperform existing models in explaining the cross-section of the 25 portfolio returns of firms sorted by size and book-to-market ratio. Conditional versions of these models perform even better, explaining as much of the crosssectional variation as even the a-theoretical Fama-French three factor model. Thus, a logical extension to Breeden s model appears to provide results that avoid most of the puzzles of the original C-CAPM. That such an extension was not attempted previously may be somewhat surprising but is at least in part due to the difficulty of dealing with the cash-in-advance lag and conditioning information in a continuous time framework. As such, the extension of Stein s lemma derived here was an important ingredient in obtaining a theoretical solution. In spite of the logic of the theory and the robustly strong explanatory power of the different versions of the model, a specification test that includes size or book-to-market ratios in addition to 27

29 our theoretical factors suggests shortcomings in the models, as well as in all other models considered here including the Fama-French three-factor model itself. Additional research is called for to propose an alternative specification that better clarifies the role of the size and book-to-market variables. 28

30 References Ahn, S. and C. Gadarowski, 2004, Small Sample Properties of the GMM Specification Test Based on the Hansen-Jagannathan Distance, Journal of Empirical Finance 11, Altonji, J. G., and Segal, L. M., 1996, Small-Sample Bias in GMM Estimation of Covariance Structures. Journal of Business and Economic Statistics 14, Balvers, R. J., T. F. Cosimano and B. McDonald, 1990, Predicting Stock Returns in an Efficient Market, Journal of Finance 45, Boyle, G.W. and L. oung, 1988, Asset Prices, Commodity Prices, and Money: A General Equilibrium, Rational Expectations Model, American Economic Review 78, Boyle, G. W. and J. D. Peterson, 1995, Monetary Policy, Aggregate Uncertainty and the Stock Market, Journal of Money, Credit and Banking 27, Breeden, D., 1979, An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities, Journal of Financial Economics 7, Brock, W. A., 1974, Money and Growth: The Case of Long Run Perfect Foresight, International Economic Review 15, Campbell, J.. and J. H. Cochrane, 1999, By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior, Journal of Political Economy 107, Chan, K.C., S. Foresi, and L. H. P. Lang, 1996, Does Money Explain Asset Returns? Theory and Empirical Analysis, Journal of Finance 51, Chen, N., R. Roll, and S. Ross, 1986, Economic Forces and the Stock Market, Journal of Business 59, Chordia, T., A. Sarkar, and A. Subrahmanyam, 2002, An Empirical Analysis of Stock and Bond Market Liquidity, Working Paper, Emory University. Cochrane, John H., 1991, Production-Based Asset Pricing and the Link Between Stock Returns and Economic Fluctuations, Journal of Finance 46, Cochrane, J. H., 1996, A Cross-Sectional Test of an Investment-based Asset Pricing Model, Journal of Political Economy 104, Cochrane, J. H., 2001, Asset Pricing, Princeton University Press, New Jersey. Daniel D. K. and S. Titman, 1997, Evidence on the Characteristics of Cross-Sectional Variation in Stock Returns, Journal of Finance, 52,

31 Fama, E. F. and K. R. French, 1992, The Cross-Section of Expected Stock Returns, Journal of Finance 47, Fama, E. F. and K. R. French, 1993, Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics 33, Fama, E. F. and K. R. French, 1996, Multifactor Explanations of Asset Pricing Anomalies, Journal of Finance 51, Fama, E. F. and J. D. MacBeth, 1973, Risk Return and Equilibrium: Empirical Tests, Journal of Political Economy 71, Feenstra, R. C., 1986, Functional Equivalence Between Liquidity Costs and the Utility of Money, Journal of Monetary Economics 17, Flannery, M. J. and A. A. Protopapadakis, 2002, Macroeconomic Factors Do Influence Aggregate Stock Returns, Review of Financial Sudies 15, Hansen, L. P. and R., K. Jagannathan, 1997, Assessing Specification Errors in Stochastic Discount Factor Models, Journal of Finance 52, Hodrick, R. and X. Zhang 2001). "Evaluating the Specification Errors of Asset Pricing Models." Journal of Financial Economics 62, Huang, C. F. and R. H. Litzenberger Foundations for Financial Economics. North Holland, New ork, N. Ingersoll, J. E Theory of Financial Decision Making. Rowman and Littlefield Publishers, Totowa, NJ. Jagannathan R. and Z. Wang, 1996, The Conditional CAPM and the Cross-Section of Expected Returns, Journal of Finance 51, Jagannathan R. and Z. Wang, 1998, An Asymptotic Theory for Estimating Beta-Pricing Models Using Cross-Sectional Regression, Journal of Finance 53, Jensen, G. R. and J. M. Mercer, 2002, Monetary Policy and the Cross-Section of Expected Stock Return, The Journal of Financial Research, XXV, Lettau, M. and S. C. Ludvigson, 2001a, Consumption, Aggregate Wealth and Expected Stock Returns, Journal of Finance 56, Lettau, M. and S. C. Ludvigson, 2001b, Resurrecting the C)CAPM: A Cross-Sectional Test When Risk Premia are Time-Varying, Journal of Political Economy 109,

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33 Appendix: A MULTIVARIATE EXTENSION OF STEIN S LEMMA In the finance literature Stein s lemma is taken to be the statement that Cov[, h X)] = E [ h X X)] σ X when X and are bivariate normal. See for instance Huang and Litzenberger 1988, pp. 101, 116) and Cochrane 2001, pp ). The lemma was first used and proven indirectly in the finance literature by Rubinstein 1973) using exact Taylor expansions. Ingersoll 1987, pp ) proves this result directly without naming it Stein s lemma. The result is generally attributed to Stein 1973, 1981). Below we prove a multivariate generalization that to our knowledge has not been proven or used elsewhere: Cov [, h X)] = n i= 1 E [ h X i X)] σ X i, where X represents a vector of multivariate random variables. This form of Stein s lemma would appear to be of great use in multi-factor asset pricing models. We first state the existing result: Stein s Lemma [Stein 1981), Lemma 2]. Assume: 1) a random variable and random n-vector ε such that and ε are independent multivariate normal, and 2) the partial derivative, ε) exists and has E g, ε) <. Then: g Cov [, g, ε )] = E [ g, ε)] σ 2 ). A1) Proof: See Stein 1981), or the following proof provided for completeness. By definition: ε Cov [, g, ε)] = g y, e) y µ ) f y, e) d y d e ε A2) 32

34 33 where y and e are realizations of the random variables and g with joined multivariate density. Note that superscripts indicate particular random variables and subscripts indicate ), e y f ε partial derivatives. Using independence and the fact that for the ) ] ) )/ [ ) 2 x f x x f X X X X X σ µ = normal distribution, we obtain A3). ) ) ), ) )],, [ 2 e e e ε ε ε d f y d y f y g g Cov = σ Integration by parts of the inner integral yields: e, A4) e e e ε ε ε d f y f y g y d y f y g g Cov ) )] ), [ ) ), ) )],, [ 2 = σ where vanishes for any e if. This is guaranteed ) ) ), e e ε f y f y g )] / [exp ) ), 2 2 y y o f y g σ = e e ε if. < ), ε g E Equation A4) then becomes, A5) e e e ε ε ε d dy f y f y g g Cov ) ) ), ) )],, [ 2 = σ which is Stein s lemma, equation A1). G Next extend the lemma to allow for dependence among the random variables.

35 Stein s Extended Lemma. Assume: 1) a random variable and random n-vector X such that and X are multivariate normal, and 2) the n-vector X) of partial h X derivatives exists with E h X X ) <. Then: Cov [, h X)] = E [ h X)] σ X X. A6) Proof: It is always possible to find coefficients b such that: 2 X = b + ε, with Cov, ε) = 0 and b = Cov, X) / σ ). A7) Then: Cov [, h X)] = h x) y µ ) f y, x) d y d x X, A8) Cov [, h X)] = h by + e) y µ ) f y, by + e) d y d ε ε, A9) Set: g, ε ) h b + ε), so that, ε) = hx b + ε) b. A10) g Using equation A10) we can apply Stein s lemma to equation A9) to yield Cov[, h X)] = E [ g, ε)] 2 σ ) = E [ hx X) ] σ X, A11) where the second equality follows from equations A7) and A10). To complete the proof note that E h X X ) < implies E X ) < and E X)' b < so that E g, ε) <, h X h X which was needed to apply Stein s lemma to A9). G 34

36 Table 1. Fama-Macbeth estimation for 6 unconditional linear factor models. The factors in each model 1 to 6: MMKT, MCCAPM, CCAPM, FF3, MKT, COCH) are market return and money supply growth Mkt, Μ2); consumption growth and money supply growth C, Μ2); consumption growth C); the Fama-French 3 factors Mkt, Smb, Hml); Market Return Mkt); and the Cochrane investment growth measures for nonresidential and residential sectors Nriq, Riq). For each model, we report the risk premiums, their t-statistic, Shanken s 1992) adjusted t-statistic, GMM t-statistics based on five lags) and the Adjusted R-square Adj. R2). Data are from 1959Q1 to 2003Q1. Model-1 Const. Mkt Μ2 Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Model-2 Const. C Μ2 Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Model-3 Const. C Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Model-4 Const. Mkt Smb Hml Adj. R2 Risk Premium % t-statistics Shanken- t Model-5 Const. Mkt Adj. R2 Risk Premium % t-statistics Shanken- t Model-6 Const. Nriq Riq Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t

37 Table 2. Fama-Macbeth estimation for 6 conditional linear factor models. The factors in each model 7 to 12) are MMKT conditioned on the lagged term premium; the MC-CAPM conditioned on the lagged term premium; the C-CAPM conditioned on the lagged term premium; the MKT conditioned on CA; the CCAPM conditioned on CA; the CCAPM conditioned on lagged consumption. For each model, we report the risk premiums, their t-statistic, Shanken s 1992) adjusted t-statistic, GMM t-statistics based on five lags) and Adj. R2. See Table 1 for descriptions of the model labels. Model-7 Const. TP Mkt )M2 TP.Mkt TP.)M2 Adj. R2 Risk Premium % t-statistics Shanken-t GMM-t Model-8 Const. TP )C )92 TP.)C IP.)92 Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Model-9 Const. TP C TP. C Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Model-10 Const. Cay Mkt Cay.Mkt Adj. R2 Risk Premium % t -statistics Shanken-t GMM-t Model-11 Const. Cay C Cay. C Adj. R2 Risk Premium % t -statistics Shanken-t GMM-t Model-12 Const. Ct 1) C Ct 1). C Adj. R2 Risk Premium % t-statistics Shanken-t GMM-t

38 Table 3. Model misspecification test incorporating the average log size or average log book-tomarket ratio. The models considered are the MMKT and MC-CAPM, the MMKT and MCCAPM conditioned on the term premium and the CA and FF3 models. See Table 1 and Table 2 for factor details. Panel A provides the results for the size variable and Panel B provides the results for the book-to-market ratio. Panel A: Size Const. Mkt Μ2 Size Adj. R2 Risk Premium % t-statistics Const. C Μ2 Size Adj. R2 Risk Premium % t-statistics Const. TP Mkt Μ2 TP.Mkt ΤP. Μ2 Size Adj. R2 Risk Premium % t-statistics Const. TP C Μ2 TP. C ΤP. Μ2 Size Adj. R2 Risk Premium % t-statistics Const. Cay Mkt Cay.Mkt Size Adj. R2 Risk Premium % t -statistics Const. Cay C Cay. C Size Adj. R2 Risk Premium % t -statistics Const. Mkt Smb Hml Size Adj. R2 Risk Premium % t-statistics

39 Table 3 continued) Panel B: Book-to-market ratio Const. Mkt Μ2 BM Adj. R2 Risk Premium % t-statistics Const. C Μ2 BM Adj. R2 Risk Premium % t-statistics Const. TP Mkt Μ2 TP.Mkt ΤP. Μ2 BM Adj. R2 Risk Premium % t-statistics Const. TP C Μ2 TP. C ΤP. Μ2 BM Adj. R2 Risk Premium % t-statistics Const. Cay Mkt Cay.Mkt BM Adj. R2 Risk Premium % t -statistics Const. Cay C Cay. C BM Adj. R2 Risk Premium % t -statistics Const. Mkt Smb Hml BM Adj. R2 Risk Premium % t-statistics

40 Table 4. Average squared pricing errors in percentages) for size and book-to-market quintiles. S1 refers to the smallest size group and B1 refers to the smallest book-to-market ratio group. The definitions for S1 and B1 extend to S2-S5 and B2-B5. The last 3 rows are the average squared pricing error, the joint test statistic for significance of the pricing errors and its p-value for the 25 size and book-to-market sorted portfolios. The pricing errors are provided for each of the six unconditional and six conditional models. See Tables 1 and 2 for model details. MMKT MCCAPM CCAPM FF3 MKT COCH S S S S S B B B B B Average Chi p-value MMKTTP MCCAPMTP CCAPMTP MKTCA CCAPMCA CCAPMC S S S S S B B B B B Average Chi p-value

41 Table 5. The distance test for linear factor pricing models using the HJ weighting matrix Hansen and Jagannathan, 1997) and Identity matrix Lettau and Ludvigson, 2001b). The test assets are the 25 portfolios sorted by size and book-to-market ratio. The distances and their statistical significance are provided for each of the six unconditional and six conditional models. See Tables 1 and 2 for model details. The p-values for the weighted P 2 statistic of Jagannathan and Wang 1996) to test for the significance of the distance measure for both weighting matrices) is based on 10,000 simulations. MMKT MCCAPM CCAPM FF3 MKT COCH HJ-Matrix p-value Identity Matrix p-value MMKTTP MCCAPMTP CCAPMTP MKTCA CCAPMCA CCAPMC HJ-Matrix p-value Identity Matrix p-value

42 Table 6. Fama-MacBeth regressions using different conditioning variables for the MCAPM and the MC-CAPM. DP is the lagged dividend-price ratio and Rf is the lagged risk free rate. Note that the conditioning variables are demeaned. See Table 1 for details on the models and variables. Const. DP Mkt Μ2 DP.Mkt DP. Μ2 Adj. R2 Risk Premium % t-statistics Shanken-t GMM-t Const. DP C Μ2 DP. C DP. Μ2 Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Const. Rf Mkt Μ2 Rf.Mkt Rf. Μ2 Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t Const. Rf C Μ2 Rf. C Rf. Μ2 Adj. R2 Risk Premium % t-statistics Shanken- t GMM-t

43 Figure 1. Actual return and predicted return for 6 unconditional linear factor models. Models 1 to 6 are MMKT, MCCAPM, CCAPM, FF3, MKT and COCH The two-digit numbers denote the Fama- French 25 portfolios. The first digit refers to the size quintile and the second digit refers to the bookto-market quintile. See Table 1 for factor details. 42

44 Figure 2. Actual return and predicted return for 6 conditional linear factor models. Models are MMKT-TP, MCCAPM-TP, CCAPM_TP, MKT-CA, CCAPM-CA and CCAPM-C The two-digit numbers denote the Fama-French 25 portfolios. The first digit refers to the size quintile and the second digit refers to the book-to-market quintile. See Table 2 for factor details. 43